How to key frame a clear action parent/child and change pivot point
A "Child Of" constraint will let you do what you want, if I've understood your question rightly.
In the constraint settings is a slider for Influence which can be key-framed (by right-clicking in the influence slider and choosing "insert keyframe"). A setting of 0 influence means the child will not follow the parent, and 1 means the parent affects it 100%.  You can have two constraints on the child so it has 2 parents - one with influence set to zero until you need it to take over influencing the child.
This constraint has location, rotation and scale axes which can be keyed by right-clicking in the box and selecting Insert Keyframe.



Edit: where it says "to to" in the picture, it should have said "go to"

Then click on those arrow heads next to there where Add Object Constraint is written, and you'll find the Child Of constraint.